1. Lecture Objectives: Answer questions related to HW 2
Write energy balance equations
Show Matlab example
Start analyzing unsteady state
Conducting

2. Homework assignment 1b Top view Te_o Te_i Tinter_surf Tair_in IDIR
Styrofoam
Te_o
Tinter_surf
Te_i
2.5 m
Tair_in
Surface
to surface
radiation
8 m
8 m
IDIR
Idif
Tw_i
South
East
Tw_o
IDIR
Tair_out
Idif
Glass
Surface to surface
radiation

3. MatLab Example Example code is posted on the class website MatLab at:

4. Conductive heat transfer
k - conductivity
of material
Steady-state
Unsteady-state
Boundary conditions
Dirichlet Tsurface = Tknown
Neumann
TS1
TS2 L h
Tair

5. Unsteady-state heat transfer (Explicit – Implicit methods)
Example:
To - known and changes in time
Tw - unknown
Ti - unknown
Ai=Ao=6 m2
(mcp)i=648 J/K
(mcp)w=9720 J/K
Initial conditions:
To = Tw = Ti = 20oC
Boundary conditions:
hi=ho=1.5 W/m2 Tw Ti To
Ao=Ai
Conservation of energy:
Time [h]
0.1
0.2
0.3
0.4
0.5
0.6
0.7 To 20 30 35 32 10 15
Time step Dt=0.1 hour = 360 s

6. Implicit methods - example
After rearranging:
2 Equations with 2 unknowns!
 = To Tw Ti
 =36 system of equation Tw Ti
 =72 system of equation Tw Ti

7. Explicit methods - example
 =360 sec
 = To Tw Ti
 =360 To Tw Ti
 =720 To Tw Ti
Time
There is NO system of equations!
UNSTABILE

8. Problems with stability !!! Often requires very small time steps
Explicit method
Problems with stability !!!
Often requires very small time steps

9. Explicit methods - example
 = To Tw Ti
 =36 To Tw Ti
 =72 To Tw Ti
Stable solution obtained
by time step reduction
10 times smaller time step
Time
 =36 sec

10. Explicit methods information progressing during the calculationTw Ti To

11. Unsteady-state conduction - Wallq Dx
Nodes for numerical
calculation

12. Discretization of a non-homogeneous wall structure
Section considered in the
following discussion
Discretization in space
Discretization in time